Abstract

The continuing rapid increase in available computing power has not reduced the importance of efficient methods of optical system assessment for automatic lens design. On the contrary, the new capabilities simply show that truly automatic optical design will eventually be accomplished. It is proposed that the merit of a system-assessment scheme be measured in terms of the accuracy of its estimation of the overall performance of a proposed system as a function of the amount of work done (e.g., number of rays traced). By using this criterion, a number of schemes based on ray tracing are compared, and some highly efficient assessment procedures are developed. As a simplifying approximation, the effects of vignetting and pupil distortion are ignored here. The key to the most-effective methods lies in coupling appropriate coordinates to Gaussian quadrature schemes. Appropriate coordinate systems are those for which the relevant integrands (either wave-front errors or transverse intercept errors) take the form of smooth functions. The resulting methods for system assessment are typically at least an order of magnitude more efficient than comparatively simple schemes.

© 1988 Optical Society of America

Full Article  |  PDF Article

Corrections

G. W. Forbes, "Optical system assessment for design: numerical ray tracing in the Gaussian pupil: erratum," J. Opt. Soc. Am. A 6, 1123-1123 (1989)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-6-8-1123

References

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  1. V. K. Viswanathan, I. O. Bohachevsky, T. P. Cotter, “An attempt to develop an ‘intelligent’ lens design program,” in 1985 International Lens Design Conference, W. H. Taylor, ed., Proc. Soc. Photo-Opt. Instrum. Eng.554, 10–17 (1985).
    [CrossRef]
  2. I. O. Bohachevsky, V. K. Viswanathan, G. Woodfin, “An ‘intelligent’ optical design program,” in Applications of Artificial Intelligence I, J. F. Gilmore, ed., Proc. Soc. Photo-Opt. Instrum. Eng.485, 104–112 (1984).
    [CrossRef]
  3. P. N. Robb, “Lens design using optical aberration coefficients,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. Soc. Photo-Opt Instrum. Eng.237, 109–118 (1980).
    [CrossRef]
  4. T. B. Andersen, “Automatic computation of optical aberration coefficients,” Appl. Opt. 19, 3800–3816 (1980).
    [CrossRef] [PubMed]
  5. G. W. Forbes, “Weighted truncation of power series and the computation of chromatic aberration coefficients,” J. Opt. Soc. Am. A 1, 350–355 (1984).
    [CrossRef]
  6. G. W. Forbes, “Extension of the convergence of multivariate aberration series,” J. Opt. Soc. Am. A 3, 1376–1383 (1986).
    [CrossRef]
  7. G. W. Forbes, “Acceleration of the convergence of multivariate aberration series,” J. Opt. Soc. Am. A 3, 1384–1394 (1986).
    [CrossRef]
  8. See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 9.1.3.
  9. This result follows on differentiation of the autocorrelation expression for the MTF. (It is interesting that the same result follows trivially for the so-called geometrical MTF from the standard relations between the moments of a function and the Taylor series of its Fourier transform.) It follows then that a system designed for minimum spot size will generally have better low-spatial-frequency response than a system designed for minimum OPD, which will have superior response to higher spatial frequencies.
  10. In the case of Gaussian quadrature, which is generally well suited to this type of problem, it is not so straightforward to design a successful method that uses the derivative, since Gaussian schemes that use both the function value and its first derivative at the sample points typically require samples out in the complex plane (even when the region of integration is a finite interval on the real axis). This is a serious drawback in this context, in which tracing rays with complex-valued field, aperture, and color variables is relatively costly. This information can be used in integration schemes of lower order, however, as shown, for example, in S. A. Comastri, J. M. Simon, “Aberration function dependence on field—a way to better obtain profit from ray tracing,” Optik 69, 135–140 (1985). The methods of polynomial fitting discussed by Comastri and Simon seem well suited to the calculation of MTF’s; however, it is not clear that such an approach is optimal for the determination of the variance of the OPD; Gaussian quadrature methods reveal that, if it is the integral that is required, it is possible to do better than fitting polynomials. For example, with two sample points, it is possible to integrate exactly any cubic over a finite interval, but only a linear function can be fitted with this data.
  11. J. W. Foreman, “Computation of rms spot radii by ray tracing,” Appl. Opt. 13, 2585–2588 (1974).
    [CrossRef] [PubMed]
  12. T. B. Andersen, “Evaluating rms spot radii by ray tracing,” Appl. Opt. 21, 1241–1248 (1982).
    [CrossRef] [PubMed]
  13. Some particular Cartesian configurations can have somewhat better performance than those quoted; however, the numbers of rays specified here are sufficient to guarantee that the error limits will typically not be violated. Although the fractional error of the Cartesian scheme dies, on average, as the inverse of the number of rays to the power 3/4, the performance is highly erratic (as one might expect with patching a square grid to a round hole), and it is difficult to guess the accuracy of a given configuration by relating it to another. For example, halving the side length of the grid (which approximately quadruples the number of rays) typically reduces the accuracy of the result for some configurations. This is discussed in detail in Section 3.
  14. G. W. Forbes, “Chromatic coordinates in aberration theory,” J. Opt. Soc. Am. A 1, 344–349 (1984).
    [CrossRef]
  15. The hypergon has a half-field angle of 65° and operates at f/30. The specifications can be found in U.S. patent706,650 (August12, 1902).
  16. The Cooke triplet used here has a half-field angle of 20° and operates at f/5.6. The specifications were taken from H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), Sec. 37, p. 60.
  17. The double Gauss used here is among the sample lens specifications provided with accos v(the lens-design program available from Scientific Calculations, Fishers, N.Y.). It has an f number of ≃2 and a half-field angle of 15°.
  18. The specifications of the Schmidt camera used here can be found in Table 3 of Ref. 4. It is designed to operate in the UV, has a half-field angle of 5°, and operates at f/1.09.
  19. The microscope objective used here was designed by J. R. Rogers of the Institute of Optics, University of Rochester. The half-cone angle at the object is 50° (numerical aperture ≈0.766), and the magnification is 50×. The specifications are available from him on request.
  20. Note that the form of the weighting functions is dependent on the choice of variables. For example, the weight for color is different if frequency is used as the coordinate in place of wavelength. If a change of variables is needed, a Jacobian must be included to find the new form of the weighting function. So, for example, in changing variables from wavelength to frequency, L(λ) is replaced by N(ν)=L[λ(ν)]dλ/dν. It is also significant that, in Eq. (2.2), a mean-square length is averaged so that if the spot size in a region near the center of the field should be weighted k times more heavily than a region near the edge, F(f) should be an extra factor of k2 higher at the center (over and above the Cartesian components of any Jacobian that is picked up by using variables other than the position vector).
  21. It is worth remarking, at this stage, that expressions of the form s2=Avgx{[f(x)−f¯]2}, which appear in Eqs. (2.6) and (2.7), are evaluated more easily if they are reformulated as follows. It is usual to expand the argument of the average operator and to reexpress s2 as s2=Avgx{f2(x)}−f¯2 in order to allow f¯ and s2 to be calculated simultaneously. However, this typically increases the numerical noise owing to cancellation. This is especially the case for the computations indicated in Eqs. (2.6) and (2.7) in which s2 may be 8 or more orders of magnitude smaller than f¯2. A more convenient expression can be obtained by first writing {f(x)−f¯} as {[f(x)−f(c)]−[f¯−f(c)]} before expanding the argument, and in this way s2=Avgx{[f(x)−f(c)]2}−[f¯−f(c)]2 is obtained, where c is taken to be some fixed value near the center of the region of interest.
  22. Sampling on a square grid is used, for example, by code v(the lens-design program available from Optical Research Associates, Pasadena, Calif.), and the polar grid can be found in sigma(the lens-design program from Kidger Optics, UK). These programs do not necessarily use the same weighting adopted here or try to calculate the same entities; I simply take these sampling schemes as a starting points for comparison purposes.
  23. If the points on a square lattice (of side length δ= R/n, where R is the radius of the disk) in one quadrant of the disk are located atrij=[δ(i−1/2),δ(j−1/2)]fori=1,2,…[(n2−1/4)1/2+1/2]andj=1,2,…[[n2−(i−1/2)2]1/2+1/2], it can be seen that the percentage error in approximating the integral of a constant over the disk by using a simple sum over these points is given byE(n)=100{S(n)−πR2}/πR2, whereS(n)=4δ2∑i[[n2−(i−1/2)2]1/2+1/2]. 〈x〉 denotes the integral part of x, and the range for the sum over i is, of course, just that indicated for the placement of points. It is remarked that scaling the overall result by a constant to ensure that all schemes integrate constant functions exactly will have no effect for the work reported in this paper, since all the integrals are for the purposes of averaging and any constant multiplying factor will cancel when the integrals are normalized to obtain the desired average. This particular sampling scheme is identical to that used in code v, and some observations are in order. This program has a parameter referred to as DEL, which is just the inverse of n. The default value of this parameter is DEL = 0.385, which corresponds to n = 2.60, which, from Fig. 3, can be seen to use 12 rays and to have an error in excess of +10% when integrating a constant (if uniform weighting is used). When the routines of Section 3 are used, this scheme is found to overestimate spot sizes consistently by 20–50%. When a value of n = 2.764 (corresponding to DEL = 0.362, the location of the zero on the plot in Fig. 3 with the same number of rays), is adopted, the error in the determination of spot size is reduced to ≃10%, an improvement by a factor of 3 to 5. This improvement is appreciated when it is recalled that the error is, on average, dropping as the number of rays to the power −3/4, so the gain realized by this minor change is equivalent to that typically obtained by increasing the number of rays by a factor of 4 to 8. For the interest of code vusers, it is noted that the locations of a number of the zeros of the curve in Fig. 3 that seem to give relatively good integration schemes are found to be n = 2.2568 (8 rays, 10–30% error), n = 2.7639 (12 rays, 4–12% error), n = 3.7424 (22 rays, 3–6% error), and n = 5.9708 (56 rays, 1–3% error).
  24. The polar scheme would probably benefit from sampling at the endpoints of the radial subdivisions rather than at the midpoints. However, there are significantly better schemes available, so this minor issue is not pursued further.
  25. With m uniformly distributed samples in [0, 2π), say, at θk= k 2π/m− γ for k= 1, 2, …, m, it can easily be shown, by using a geometric series, that Σk cos qθk vanishes for q not equal to a multiple of m. Now, since cospθ can be written as a linear combination of {cos qθ; q= p, p− 2, p− 4, …, 0 or 1}, it follows that uniform weighting with m uniformly distributed points can be used to integrate exactly cospθ for p= 0, 1, 2, …, m− 1. The use of alternating weights can be regarded as a superposition of a scheme with 2n points, and one with only half that number (skipping every other one) and can be used to integrate exactly cospθ for p= 1, 2, …, n− 1, where uniform weights yield exact results for p= 1, 2, …, 2n− 1.
  26. This result also holds for the sample points θj= jπ/Nθ, j= 0, 1, …, Nθ, which includes points on the line of symmetry (i.e., meridional rays), although the same accuracy is now obtained with one extra point on each ring.
  27. For interest, it is remarked that since the Legendre polynomials are simply related to the rotationally symmetric Zernike polynomials, it can be seen that the sample points presented in Table 1 in fact correspond to the radial locations of the zeros of these Zernike polynomials. The values of the weights follow simply from the relation between Gaussian quadrature and orthogonal polynomials, which is presented in H. S. Wilf, Mathematics for the Physical Sciences (Dover, New York, 1962), Sec. 2.9, pp. 61–64. A convenient form of the recurrence relations for generating the orthogonal polynomials from which the parameters for Gaussian quadrature can be found is presented in W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Sec. 4.5.
  28. For a simple description of the derivation of the Radau integration methods, see, for example, R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1973), Sec. 19.7, pp. 328–330.
  29. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968) Sec. 87, pp. 150–154.
  30. See, for example, P. N. Robb, “Selection of optical glasses. I: Two materials,” Appl. Opt. 24, 1864–1877 (1985).
    [CrossRef] [PubMed]
  31. J. R. Rogers, M. D. Hopler, “Conversion of group refractive index to phase refractive index,” J. Opt. Soc. Am. A 5, 1595–1600 (1988).
    [CrossRef]
  32. This somewhat vague expectation can be given a more definite meaning and verified as follows. The monic polynomial of degree m that has the smallest mean-square value over a given interval can be shown to be proportional to the m th-order orthogonal polynomial over that interval, say, ϕm. A Gaussian quadrature scheme with m sample points has roots at the locations of the zeros of this very polynomial. Such an integration scheme is clearly unable to determine the mean-square value of ϕm: it exactly integrates polynomials of degree less than or equal to 2m− 1, whereas the square of ϕm is of degree 2m. Nevertheless, if used as a merit function, the Gaussian integration scheme reports a mean-square value of zero for an m th-order polynomial if and only if the polynomial is a multiple of ϕm. This means that, in the sense of minimizing the mean-square value, the m th-order term is optimally balanced by the terms of lower order. In practice, this entails that when the Gaussian merit functions discussed in this paper seriously underestimate, say, a mean-square wave aberration owing to the dominance of aberrations of higher order than those that the scheme can account for, the balancing of the unseen terms will not be far from optimal. In this sense, the toothpaste tube is being squeezed at just those points that guarantee that the smallest possible volume is left inside in the event that the thickness at each point is reduced to zero.

1988 (1)

1986 (2)

1985 (2)

In the case of Gaussian quadrature, which is generally well suited to this type of problem, it is not so straightforward to design a successful method that uses the derivative, since Gaussian schemes that use both the function value and its first derivative at the sample points typically require samples out in the complex plane (even when the region of integration is a finite interval on the real axis). This is a serious drawback in this context, in which tracing rays with complex-valued field, aperture, and color variables is relatively costly. This information can be used in integration schemes of lower order, however, as shown, for example, in S. A. Comastri, J. M. Simon, “Aberration function dependence on field—a way to better obtain profit from ray tracing,” Optik 69, 135–140 (1985). The methods of polynomial fitting discussed by Comastri and Simon seem well suited to the calculation of MTF’s; however, it is not clear that such an approach is optimal for the determination of the variance of the OPD; Gaussian quadrature methods reveal that, if it is the integral that is required, it is possible to do better than fitting polynomials. For example, with two sample points, it is possible to integrate exactly any cubic over a finite interval, but only a linear function can be fitted with this data.

See, for example, P. N. Robb, “Selection of optical glasses. I: Two materials,” Appl. Opt. 24, 1864–1877 (1985).
[CrossRef] [PubMed]

1984 (2)

1982 (1)

1980 (1)

1974 (1)

Andersen, T. B.

Bohachevsky, I. O.

V. K. Viswanathan, I. O. Bohachevsky, T. P. Cotter, “An attempt to develop an ‘intelligent’ lens design program,” in 1985 International Lens Design Conference, W. H. Taylor, ed., Proc. Soc. Photo-Opt. Instrum. Eng.554, 10–17 (1985).
[CrossRef]

I. O. Bohachevsky, V. K. Viswanathan, G. Woodfin, “An ‘intelligent’ optical design program,” in Applications of Artificial Intelligence I, J. F. Gilmore, ed., Proc. Soc. Photo-Opt. Instrum. Eng.485, 104–112 (1984).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 9.1.3.

Buchdahl, H. A.

The Cooke triplet used here has a half-field angle of 20° and operates at f/5.6. The specifications were taken from H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), Sec. 37, p. 60.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968) Sec. 87, pp. 150–154.

Comastri, S. A.

In the case of Gaussian quadrature, which is generally well suited to this type of problem, it is not so straightforward to design a successful method that uses the derivative, since Gaussian schemes that use both the function value and its first derivative at the sample points typically require samples out in the complex plane (even when the region of integration is a finite interval on the real axis). This is a serious drawback in this context, in which tracing rays with complex-valued field, aperture, and color variables is relatively costly. This information can be used in integration schemes of lower order, however, as shown, for example, in S. A. Comastri, J. M. Simon, “Aberration function dependence on field—a way to better obtain profit from ray tracing,” Optik 69, 135–140 (1985). The methods of polynomial fitting discussed by Comastri and Simon seem well suited to the calculation of MTF’s; however, it is not clear that such an approach is optimal for the determination of the variance of the OPD; Gaussian quadrature methods reveal that, if it is the integral that is required, it is possible to do better than fitting polynomials. For example, with two sample points, it is possible to integrate exactly any cubic over a finite interval, but only a linear function can be fitted with this data.

Cotter, T. P.

V. K. Viswanathan, I. O. Bohachevsky, T. P. Cotter, “An attempt to develop an ‘intelligent’ lens design program,” in 1985 International Lens Design Conference, W. H. Taylor, ed., Proc. Soc. Photo-Opt. Instrum. Eng.554, 10–17 (1985).
[CrossRef]

Forbes, G. W.

Foreman, J. W.

Hamming, R. W.

For a simple description of the derivation of the Radau integration methods, see, for example, R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1973), Sec. 19.7, pp. 328–330.

Hopler, M. D.

Robb, P. N.

See, for example, P. N. Robb, “Selection of optical glasses. I: Two materials,” Appl. Opt. 24, 1864–1877 (1985).
[CrossRef] [PubMed]

P. N. Robb, “Lens design using optical aberration coefficients,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. Soc. Photo-Opt Instrum. Eng.237, 109–118 (1980).
[CrossRef]

Rogers, J. R.

Simon, J. M.

In the case of Gaussian quadrature, which is generally well suited to this type of problem, it is not so straightforward to design a successful method that uses the derivative, since Gaussian schemes that use both the function value and its first derivative at the sample points typically require samples out in the complex plane (even when the region of integration is a finite interval on the real axis). This is a serious drawback in this context, in which tracing rays with complex-valued field, aperture, and color variables is relatively costly. This information can be used in integration schemes of lower order, however, as shown, for example, in S. A. Comastri, J. M. Simon, “Aberration function dependence on field—a way to better obtain profit from ray tracing,” Optik 69, 135–140 (1985). The methods of polynomial fitting discussed by Comastri and Simon seem well suited to the calculation of MTF’s; however, it is not clear that such an approach is optimal for the determination of the variance of the OPD; Gaussian quadrature methods reveal that, if it is the integral that is required, it is possible to do better than fitting polynomials. For example, with two sample points, it is possible to integrate exactly any cubic over a finite interval, but only a linear function can be fitted with this data.

Viswanathan, V. K.

V. K. Viswanathan, I. O. Bohachevsky, T. P. Cotter, “An attempt to develop an ‘intelligent’ lens design program,” in 1985 International Lens Design Conference, W. H. Taylor, ed., Proc. Soc. Photo-Opt. Instrum. Eng.554, 10–17 (1985).
[CrossRef]

I. O. Bohachevsky, V. K. Viswanathan, G. Woodfin, “An ‘intelligent’ optical design program,” in Applications of Artificial Intelligence I, J. F. Gilmore, ed., Proc. Soc. Photo-Opt. Instrum. Eng.485, 104–112 (1984).
[CrossRef]

Wilf, H. S.

For interest, it is remarked that since the Legendre polynomials are simply related to the rotationally symmetric Zernike polynomials, it can be seen that the sample points presented in Table 1 in fact correspond to the radial locations of the zeros of these Zernike polynomials. The values of the weights follow simply from the relation between Gaussian quadrature and orthogonal polynomials, which is presented in H. S. Wilf, Mathematics for the Physical Sciences (Dover, New York, 1962), Sec. 2.9, pp. 61–64. A convenient form of the recurrence relations for generating the orthogonal polynomials from which the parameters for Gaussian quadrature can be found is presented in W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Sec. 4.5.

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 9.1.3.

Woodfin, G.

I. O. Bohachevsky, V. K. Viswanathan, G. Woodfin, “An ‘intelligent’ optical design program,” in Applications of Artificial Intelligence I, J. F. Gilmore, ed., Proc. Soc. Photo-Opt. Instrum. Eng.485, 104–112 (1984).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. A (5)

Optik (1)

In the case of Gaussian quadrature, which is generally well suited to this type of problem, it is not so straightforward to design a successful method that uses the derivative, since Gaussian schemes that use both the function value and its first derivative at the sample points typically require samples out in the complex plane (even when the region of integration is a finite interval on the real axis). This is a serious drawback in this context, in which tracing rays with complex-valued field, aperture, and color variables is relatively costly. This information can be used in integration schemes of lower order, however, as shown, for example, in S. A. Comastri, J. M. Simon, “Aberration function dependence on field—a way to better obtain profit from ray tracing,” Optik 69, 135–140 (1985). The methods of polynomial fitting discussed by Comastri and Simon seem well suited to the calculation of MTF’s; however, it is not clear that such an approach is optimal for the determination of the variance of the OPD; Gaussian quadrature methods reveal that, if it is the integral that is required, it is possible to do better than fitting polynomials. For example, with two sample points, it is possible to integrate exactly any cubic over a finite interval, but only a linear function can be fitted with this data.

Other (22)

This somewhat vague expectation can be given a more definite meaning and verified as follows. The monic polynomial of degree m that has the smallest mean-square value over a given interval can be shown to be proportional to the m th-order orthogonal polynomial over that interval, say, ϕm. A Gaussian quadrature scheme with m sample points has roots at the locations of the zeros of this very polynomial. Such an integration scheme is clearly unable to determine the mean-square value of ϕm: it exactly integrates polynomials of degree less than or equal to 2m− 1, whereas the square of ϕm is of degree 2m. Nevertheless, if used as a merit function, the Gaussian integration scheme reports a mean-square value of zero for an m th-order polynomial if and only if the polynomial is a multiple of ϕm. This means that, in the sense of minimizing the mean-square value, the m th-order term is optimally balanced by the terms of lower order. In practice, this entails that when the Gaussian merit functions discussed in this paper seriously underestimate, say, a mean-square wave aberration owing to the dominance of aberrations of higher order than those that the scheme can account for, the balancing of the unseen terms will not be far from optimal. In this sense, the toothpaste tube is being squeezed at just those points that guarantee that the smallest possible volume is left inside in the event that the thickness at each point is reduced to zero.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 9.1.3.

This result follows on differentiation of the autocorrelation expression for the MTF. (It is interesting that the same result follows trivially for the so-called geometrical MTF from the standard relations between the moments of a function and the Taylor series of its Fourier transform.) It follows then that a system designed for minimum spot size will generally have better low-spatial-frequency response than a system designed for minimum OPD, which will have superior response to higher spatial frequencies.

Some particular Cartesian configurations can have somewhat better performance than those quoted; however, the numbers of rays specified here are sufficient to guarantee that the error limits will typically not be violated. Although the fractional error of the Cartesian scheme dies, on average, as the inverse of the number of rays to the power 3/4, the performance is highly erratic (as one might expect with patching a square grid to a round hole), and it is difficult to guess the accuracy of a given configuration by relating it to another. For example, halving the side length of the grid (which approximately quadruples the number of rays) typically reduces the accuracy of the result for some configurations. This is discussed in detail in Section 3.

V. K. Viswanathan, I. O. Bohachevsky, T. P. Cotter, “An attempt to develop an ‘intelligent’ lens design program,” in 1985 International Lens Design Conference, W. H. Taylor, ed., Proc. Soc. Photo-Opt. Instrum. Eng.554, 10–17 (1985).
[CrossRef]

I. O. Bohachevsky, V. K. Viswanathan, G. Woodfin, “An ‘intelligent’ optical design program,” in Applications of Artificial Intelligence I, J. F. Gilmore, ed., Proc. Soc. Photo-Opt. Instrum. Eng.485, 104–112 (1984).
[CrossRef]

P. N. Robb, “Lens design using optical aberration coefficients,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. Soc. Photo-Opt Instrum. Eng.237, 109–118 (1980).
[CrossRef]

The hypergon has a half-field angle of 65° and operates at f/30. The specifications can be found in U.S. patent706,650 (August12, 1902).

The Cooke triplet used here has a half-field angle of 20° and operates at f/5.6. The specifications were taken from H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), Sec. 37, p. 60.

The double Gauss used here is among the sample lens specifications provided with accos v(the lens-design program available from Scientific Calculations, Fishers, N.Y.). It has an f number of ≃2 and a half-field angle of 15°.

The specifications of the Schmidt camera used here can be found in Table 3 of Ref. 4. It is designed to operate in the UV, has a half-field angle of 5°, and operates at f/1.09.

The microscope objective used here was designed by J. R. Rogers of the Institute of Optics, University of Rochester. The half-cone angle at the object is 50° (numerical aperture ≈0.766), and the magnification is 50×. The specifications are available from him on request.

Note that the form of the weighting functions is dependent on the choice of variables. For example, the weight for color is different if frequency is used as the coordinate in place of wavelength. If a change of variables is needed, a Jacobian must be included to find the new form of the weighting function. So, for example, in changing variables from wavelength to frequency, L(λ) is replaced by N(ν)=L[λ(ν)]dλ/dν. It is also significant that, in Eq. (2.2), a mean-square length is averaged so that if the spot size in a region near the center of the field should be weighted k times more heavily than a region near the edge, F(f) should be an extra factor of k2 higher at the center (over and above the Cartesian components of any Jacobian that is picked up by using variables other than the position vector).

It is worth remarking, at this stage, that expressions of the form s2=Avgx{[f(x)−f¯]2}, which appear in Eqs. (2.6) and (2.7), are evaluated more easily if they are reformulated as follows. It is usual to expand the argument of the average operator and to reexpress s2 as s2=Avgx{f2(x)}−f¯2 in order to allow f¯ and s2 to be calculated simultaneously. However, this typically increases the numerical noise owing to cancellation. This is especially the case for the computations indicated in Eqs. (2.6) and (2.7) in which s2 may be 8 or more orders of magnitude smaller than f¯2. A more convenient expression can be obtained by first writing {f(x)−f¯} as {[f(x)−f(c)]−[f¯−f(c)]} before expanding the argument, and in this way s2=Avgx{[f(x)−f(c)]2}−[f¯−f(c)]2 is obtained, where c is taken to be some fixed value near the center of the region of interest.

Sampling on a square grid is used, for example, by code v(the lens-design program available from Optical Research Associates, Pasadena, Calif.), and the polar grid can be found in sigma(the lens-design program from Kidger Optics, UK). These programs do not necessarily use the same weighting adopted here or try to calculate the same entities; I simply take these sampling schemes as a starting points for comparison purposes.

If the points on a square lattice (of side length δ= R/n, where R is the radius of the disk) in one quadrant of the disk are located atrij=[δ(i−1/2),δ(j−1/2)]fori=1,2,…[(n2−1/4)1/2+1/2]andj=1,2,…[[n2−(i−1/2)2]1/2+1/2], it can be seen that the percentage error in approximating the integral of a constant over the disk by using a simple sum over these points is given byE(n)=100{S(n)−πR2}/πR2, whereS(n)=4δ2∑i[[n2−(i−1/2)2]1/2+1/2]. 〈x〉 denotes the integral part of x, and the range for the sum over i is, of course, just that indicated for the placement of points. It is remarked that scaling the overall result by a constant to ensure that all schemes integrate constant functions exactly will have no effect for the work reported in this paper, since all the integrals are for the purposes of averaging and any constant multiplying factor will cancel when the integrals are normalized to obtain the desired average. This particular sampling scheme is identical to that used in code v, and some observations are in order. This program has a parameter referred to as DEL, which is just the inverse of n. The default value of this parameter is DEL = 0.385, which corresponds to n = 2.60, which, from Fig. 3, can be seen to use 12 rays and to have an error in excess of +10% when integrating a constant (if uniform weighting is used). When the routines of Section 3 are used, this scheme is found to overestimate spot sizes consistently by 20–50%. When a value of n = 2.764 (corresponding to DEL = 0.362, the location of the zero on the plot in Fig. 3 with the same number of rays), is adopted, the error in the determination of spot size is reduced to ≃10%, an improvement by a factor of 3 to 5. This improvement is appreciated when it is recalled that the error is, on average, dropping as the number of rays to the power −3/4, so the gain realized by this minor change is equivalent to that typically obtained by increasing the number of rays by a factor of 4 to 8. For the interest of code vusers, it is noted that the locations of a number of the zeros of the curve in Fig. 3 that seem to give relatively good integration schemes are found to be n = 2.2568 (8 rays, 10–30% error), n = 2.7639 (12 rays, 4–12% error), n = 3.7424 (22 rays, 3–6% error), and n = 5.9708 (56 rays, 1–3% error).

The polar scheme would probably benefit from sampling at the endpoints of the radial subdivisions rather than at the midpoints. However, there are significantly better schemes available, so this minor issue is not pursued further.

With m uniformly distributed samples in [0, 2π), say, at θk= k 2π/m− γ for k= 1, 2, …, m, it can easily be shown, by using a geometric series, that Σk cos qθk vanishes for q not equal to a multiple of m. Now, since cospθ can be written as a linear combination of {cos qθ; q= p, p− 2, p− 4, …, 0 or 1}, it follows that uniform weighting with m uniformly distributed points can be used to integrate exactly cospθ for p= 0, 1, 2, …, m− 1. The use of alternating weights can be regarded as a superposition of a scheme with 2n points, and one with only half that number (skipping every other one) and can be used to integrate exactly cospθ for p= 1, 2, …, n− 1, where uniform weights yield exact results for p= 1, 2, …, 2n− 1.

This result also holds for the sample points θj= jπ/Nθ, j= 0, 1, …, Nθ, which includes points on the line of symmetry (i.e., meridional rays), although the same accuracy is now obtained with one extra point on each ring.

For interest, it is remarked that since the Legendre polynomials are simply related to the rotationally symmetric Zernike polynomials, it can be seen that the sample points presented in Table 1 in fact correspond to the radial locations of the zeros of these Zernike polynomials. The values of the weights follow simply from the relation between Gaussian quadrature and orthogonal polynomials, which is presented in H. S. Wilf, Mathematics for the Physical Sciences (Dover, New York, 1962), Sec. 2.9, pp. 61–64. A convenient form of the recurrence relations for generating the orthogonal polynomials from which the parameters for Gaussian quadrature can be found is presented in W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Sec. 4.5.

For a simple description of the derivation of the Radau integration methods, see, for example, R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1973), Sec. 19.7, pp. 328–330.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968) Sec. 87, pp. 150–154.

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Figures (14)

Fig. 1
Fig. 1

Example of Cartesian sampling in the pupil. With uniform weighting, this particular grid typically results in 3% accuracy in the calculation of spot size with 26 rays, which is shown to be one of the more-efficient Cartesian schemes.

Fig. 2
Fig. 2

Example of a uniform polar grid. The sample points are arranged on equally spaced rings with a fixed arc length separating the points on each ring. With uniform weighting, this particular grid yields about 15% accuracy in the calculation of spot size with 16 rays.

Fig. 3
Fig. 3

Efficiency of numerically integrating to determine the area of a disk by using a Cartesian grid with uniform weighting. The downward spikes in the error curve represent continuous changes of sign and locate configurations that exactly integrate a constant over the disk. The turnover points, where the error is at a local maximum, occur when new sample points cross over the edge of the disk (as can be seen by correlating the two curves), and this causes the error to jump discontinuously from a negative value to a positive one.

Fig. 4
Fig. 4

Efficiencies of a variety of integration schemes for the determination of spot size. The curves are plotted for (■) a Cartesian grid with uniform weights; (□) a polar grid with uniform weights; (▲) Andersen’s scheme, which is effectively Simpson’s method on a Cartesian grid in the polar coordinate plane (r, θ), where the disk takes the form of a rectangle; (Δ) a modified version of Andersen’s method that uses uniform weighting for the angular integral; and (♦) a scheme that uses Gaussian quadrature in the radial direction. The curves represent typical errors. By chance, a given method may have significantly better accuracy on some occasions, but, for the systems analyzed here, the errors rarely exceeded those reported by more than a factor of 2 or so (a relatively small shift on this log plot).

Fig. 5
Fig. 5

An example of the type of grid used by Andersen to sample the pupil. With Andersen’s weighting, this particular grid yields ≃10% accuracy with 21 rays.

Fig. 6
Fig. 6

Comparison of the efficiencies of schemes based on the radial coordinate r with those based on the coordinate ρ, which is just a normalized version of r2. The first three methods in the legend are described in Fig. 4. The latter three are based on ρ and include (■) a trapezoidal scheme, (×) a fourth-order scheme, and (□) Gaussian quadrature, respectively.

Fig. 7
Fig. 7

Sample points for the nine-ray configuration based on Gaussian quadrature. This scheme (when the weights in Table 1 are used for Nr = 3) has an accuracy of ≃1%.

Fig. 8
Fig. 8

Seven-ray configuration, with a Radau method used for the radial integral. This sampling scheme, together with the weights from Table 2, for Nr = 2, yields ≃3% accuracy for calculating spot-size information.

Fig. 9
Fig. 9

Relative spectral weight adopted as an example for the color average.

Fig. 10
Fig. 10

Comparison of the efficiencies of a number of schemes for averaging over color. The schemes include (▲) uniform sampling in wavelength; (♦) Gaussian quadrature, assuming that the integrand is approximated well by a polynomial in wavelength; and (■) Gaussian quadrature, assuming that the integrand is approximated well by a polynomial in Buchdahl’s chromatic coordinate.

Fig. 11
Fig. 11

Chromatic dependence of the spot size and the centroid location for the double gauss halfway out in the field. (The unit of length is chosen to be 1/100 of the focal length.) The plane curves are the quadratic and the quartic of the best fits to the upper and lower curves, respectively, with wavelength used as the variable. Note that these curves can be fitted so closely by a quadratic and a quartic, respectively, in Buchdahl’s chromatic coordinate that the results are indistinguishable from the originals on this scale.

Fig. 12
Fig. 12

Weight function, adopted as an example for averaging the mean-square spot size over the field. The function is of the form W(h) = 1.0 − a1h2a2h4, where h is the normalized object position and the coefficients a1 and a2 are taken to have the values 0.25 and 0.5, respectively.

Fig. 13
Fig. 13

Comparison of the better methods for averaging over the field. The first method involves uniform sampling in the object position (or direction tangents) squared and corresponds to the trapezoidal scheme used for the aperture average (although here the samples are taken to be the midpoints of the subintervals). The other two methods involve Gaussian and Radau quadratures of the same form as the most-efficient methods developed for the aperture average.

Fig. 14
Fig. 14

Efficiencies of a number of global averaging schemes (determined from the results of Sections 3–5) are compared in this plot. Curves: □, Cartesian sampling in the pupil, a Gaussian method based on wavelength for the color average and the trapezoidal rule based on the square of the field variable for the field average; ■, a similar method, except for the aperture average, for which Andersen’s method is used (these two curves serve as starting points by which any alternative scheme can be measured); ▲, a trapezoidal scheme based on the square of the aperture variable and Gaussian methods in the color and field averaging based on Buchdahl’s chromatic coordinate and the square of the field variable, respectively (this curve is expected to give some indication of the limits of numerical ray tracing when vignetting plays a significant role and is to be accounted for directly); ×, the most-effective Gaussian schemes for each average.

Tables (9)

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Table 1 Gaussian Integration Parameters for the Radial Integral as Approximated in Eq. (3.5)

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Table 2 Radau Integration Parameters for the Radial Integral as Approximated in Eq. (3.6)

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Table 3 Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.2 (i.e., U ≈ 27°)

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Table 4 Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.4 (i.e., U ≈ 39°)

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Table 5 Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.6 (i.e., U ≈ 51°)

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Table 6 Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.8 (i.e., U ≈ 63°)

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Table 7 Gaussian Integration Parameters for Integration over the Visible Spectrum with Uniform Weight in Wavelength and with the Gaussian Weight Plotted in Fig. 9a

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Table 8 Gaussian Integration Parameters for Averaging over the Field by Using the Weight Function Presented in Fig. 12

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Table 9 Radau Integration Parameters for Averaging over the Field by Using the Weight Function Presented in Fig. 12

Equations (21)

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Av g x { f ( x ) } : = f ( x ) χ ( x ) d x χ ( x ) d x ,
M 2 : = Av g f { L 2 ( f ) } .
Y ¯ ( f , ω ) : = Av g a { Y ( f , a , ω ) } .
Y ¯ ¯ ( f ) = Av g ω { Y ¯ ( f , ω ) } = Av g a , ω { Y ( f , a , ω ) } .
L 2 ( f ) : = Av g a , ω { [ Y ( f , a , ω ) Y ¯ ¯ ( f ) ] 2 } = Av g a , ω { ( [ Y ( f , a , ω ) Y ¯ ( f , ω ) ] + [ Y ¯ ( f , ω ) Y ¯ ( f ) ] ) 2 } = Av g ω { Av g a { [ Y ( f , a , ω ) Y ¯ ( f , ω ) ] 2 } + [ Y ¯ ( f , ω ) Y ¯ ( f ) ] 2 } .
σ 2 ( f , ω ) : = Av g a { [ Y ( f , a , ω ) Y ¯ ( f , ω ) ] 2 } ,
L 2 ( f ) = Av g ω { σ 2 ( f , ω ) + [ Y ¯ ( f , ω ) Y ¯ ¯ ( f ) ] 2 } .
σ 2 ( f , ω ) = S 0 ( f , ω ) + 2 Δ S 1 ( f , ω ) + Δ 2 S 2 ( f , ω ) ,
L 2 ( f ) = L 0 ( f ) + 2 Δ L 1 ( f ) + Δ 2 L 2 ( f ) ,
M 2 = M 0 + 2 Δ M 1 + Δ 2 M 2 .
0 R f ( r , θ ) r d r = R 2 0 1 f ( R u , θ ) u d u R 2 j = 1 N r w j f ( R u j , θ ) ,
I : = 0 R 0 2 π f ( r , θ ) d θ r d r = R 2 0 1 { 2 0 π f ( R u , θ ) d θ } u d u R 2 j = 1 N r w j { 2 π N θ k = 1 N θ f ( R u j , θ k ) } ,
I = R 2 0 1 { 2 0 π f ( R ρ 1 / 2 , θ ) d θ } d ρ 2
1 2 R 2 j = 0 N r w j { 2 π N θ k = 1 N θ f [ R ( j N r ) 1 / 2 , θ k ] } ,
I R 2 j = 1 N r w j { 2 π N θ k = 1 N θ f ( R ρ j 1 / 2 , θ k ) }
I R 2 j = 0 N r υ j { 2 π N θ k = 1 N θ f ( R u j , θ k ) } ,
0 R f ( r , θ ) r d r = D 2 0 S f [ D ( ξ 1 ξ ) 1 / 2 , θ ] d ξ 2 ( 1 ξ ) 2 ,
I = R 2 0 I { 2 0 π f ( R [ ( 1 S ) υ 1 S υ ] 1 / 2 , θ ) d θ } 1 S 2 ( 1 S υ ) 2 d υ R 2 j = 1 N r x j { 2 π N θ k = 1 N θ f ( R s j , θ k ) } .
rij=[δ(i1/2),δ(j1/2)]fori=1,2,[(n21/4)1/2+1/2]andj=1,2,[[n2(i1/2)2]1/2+1/2],
E(n)=100{S(n)πR2}/πR2,
S(n)=4δ2i[[n2(i1/2)2]1/2+1/2].

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